Problem 7 - Calculus Concepts

Question

Swim Time The time it takes an average athlete to swim 100 meters freestyle at age \(x\) years can be modeled as \(t(x)=0.181 x^{2}-8.463 x+147.376\) seconds (Source: Based on data from Swimming World, August 1992 ) a. Numerically estimate to the nearest tenth the rate of change of the time for a 13 -year-old swimmer to swim 100 meters freestyle. b. Determine the percentage rate of change of swim time for a 13-year-old. c. Is a 13-year-old swimmer's time improving or getting worse as the swimmer gets older?

Step-by-Step Solution

Answer

a) -3.8 seconds/year. b) -5.6%. c) Improving.

Step 1: Identify the rate of change formula

The rate of change of a function at a particular point can be estimated using the derivative. Here, the function given is \(t(x) = 0.181x^2 - 8.463x + 147.376\). We need the derivative \(t'(x)\) to estimate the rate of change at \(x = 13\).

Step 2: Differentiate the function

To find the derivative \(t'(x)\), apply the rules of differentiation: \[t'(x) = \frac{d}{dx}(0.181x^2) - \frac{d}{dx}(8.463x) + \frac{d}{dx}(147.376).\]This simplifies to:\[t'(x) = 2 \times 0.181x - 8.463.\]Thus,\[t'(x) = 0.362x - 8.463.\]

Step 3: Evaluate the derivative at x = 13

Substitute \(x = 13\) into the derivative to find the rate of change at that age:\[t'(13) = 0.362 \times 13 - 8.463 = 4.706 - 8.463 \approx -3.757.\]Hence, the rate of change is approximately \(-3.8\) seconds/year.

Step 4: Find the initial swim time at age 13

Substitute \(x = 13\) into the original time function to find the swimming time:\[t(13) = 0.181 \times 13^2 - 8.463 \times 13 + 147.376.\]After calculation,\[t(13) = 30.617 - 110.019 + 147.376 = 67.974.\]So, the swim time at age 13 is approximately 68.0 seconds.

Step 5: Calculate the percentage rate of change

The percentage rate of change is given by:\[\text{Percentage Rate of Change} = \left(\frac{\text{Rate of Change}}{\text{Initial Value}}\right) \times 100\%\]Substitute the known values:\[\left(\frac{-3.8}{68.0}\right) \times 100\% \approx -5.6\%.\]

Step 6: Interpret the results

Since the rate of change is negative, it indicates that the time is decreasing with age for a 13-year-old swimmer. This means the swimmer's performance is improving, as it takes less time to swim 100 meters as they get older.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus used to determine how a function changes at any given point on its curve. In simpler terms, differentiation tells us the rate at which a function's output changes as its input changes.

This is crucial for understanding things like speed, growth, and efficiency in various fields, from physics to economics. The process of finding a derivative involves applying some established rules to the function.For instance, in the original problem, we applied differentiation to a quadratic function, which was given by:

• The original function is: \( t(x) = 0.181x^2 - 8.463x + 147.376 \)
• The derivative, \( t'(x) \), is calculated to be: \( 0.362x - 8.463 \)

Differentiation helps us find this derivative, which tells us how fast the swim time is changing relative to age. By substituting \( x = 13 \) into the derivative we calculated \( t'(13) \) to understand the change specifically for a 13-year-old athlete.

Percentage Rate of Change

The percentage rate of change is a measure that allows us to understand the relative change between two quantities. It provides insight on how much one quantity changes in terms of percentage, which is often more intuitive and relatable.To determine the percentage rate of change, we use this formula:

• \( \text{Percentage Rate of Change} = \left( \frac{\text{Rate of Change}}{\text{Initial Value}} \right) \times 100\% \)

In our example, we derived a rate of change, \(-3.8\) seconds per year, for the 13-year-old swimmer's time. By plugging it into our formula together with the initial speed of \(68.0\) seconds:

• Result: \(-5.6\%\)

A negative percentage indicates that as the athlete ages, their swim time decreases, reflecting an improvement in their performance. The time taken to swim 100 meters gets shorter, thus showing better efficiency and speed.

Mathematical Modeling

Mathematical modeling involves creating a mathematical representation of a real-world scenario. This can help in predicting future trends and understanding existing patterns.In our case, the function \( t(x) = 0.181x^2 - 8.463x + 147.376 \) is a mathematical model representing the time it takes for an average athlete to swim 100 meters based on their age. This type of model can guide predictions and assumptions in athletic performance.Creating these models requires assembling data from observations and using mathematical structures such as functions to describe these relationships.

• They allow decision-makers to simulate different scenarios and analyze potential outcomes.

By using the model, coaches can make informed decisions on training regimens, pinpoint times for peak performances, and continue evaluating strategies for improvement. It's a potent tool for making predictions about a swimmer's future performance as well as understanding the underlying improvements as they grow older.